Multi-probe linear fitting and time of arrival linear correction method to analyze blade vibration based on blade tip timing without once-per-revolution

Snqun REN, Xiorong XIANG, Qingjun ZHAO,c,*, Weimin WANG,Wei ZHAO,c, Chunlin ZHAN

a Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China

b School of Aeronautics and Astronautics, University of Chinese Academy of Sciences, Beijing 100049, China

c Innovation Academy for Light-duty Gas Turbine, Chinese Academy of Sciences, Beijing 100190, China

d Beijing University of Chemical Technology, Beijing 100029, China

KEYWORDS

Abstract Blade vibration monitoring can ensure the safe operation of aero-engine rotor blades.Among the methods of blade vibration monitoring,Blade Tip Timing (BTT)method has attracted more and more attention because of its advantages of non-contact measurement.However,it is difficult to install the Once-Per-Revolution(OPR)probe in the confined space of aero-engine,and the failure and instability of the OPR signal will reduce the reliability of the blade vibration analysis results, which directly affects the accuracy of the blade vibration parameters identification. The Multi-Probe linear fitting and Time of Arrival (ToA) Linear Correction method based on the BTT (MP-LC-BTT) without OPR is proposed to reduce the errors of single probe linear fitting method for blade vibration displacement analysis.The proposed method can also correct the calculation error of blade vibration displacement due to the nonlinear change of rotation speed, which can improve the analysis accuracy of the blade vibration displacement. A new blade vibration model conforming to the actual vibration characteristics is established, and the effectiveness of the proposed method is verified by numerical simulation. Finally, the reliability and accuracy of the MP-LC-BTT method have been verified by the experiments which include two high-speed blade test-benches and an industrial axial fan. This method can be used in the actual aero-engine monitoring instead of the BTT method with OPR.

1. Introduction

Compared with the traditional method of sticking strain gauge on the blade surface,the Blade Tip Timing(BTT) method has the advantage of being able to measure the whole-circle blades at the same time.1,2The strain gauge method will change the shape of the blade surface, which will interfere with the organization ability of the flow field of the rotor blades and affect the aerodynamic performance of the engine. However, the BTT method is non-intrusive and will not.For the blade vibration monitoring of aero-engine during ground test or flight,the strain gauge method cannot be used, so the BTT method has been concerned in aero-engine blade vibration monitoring. In recent years, many scholars have produced a lot of literature on blade vibration measurement and blade vibration parameters identification, which has promoted the engineering availability of BTT method.3-5

The main research directions based on BTT method are divided into three aspects.The first is the research on the blade vibration monitoring method and the vibration displacement analysis method;6,7the second is the research on the blade vibration parameters identification;8,9the third is the application of the obtained blade vibration parameters,such as blade crack identification10and blade vibration stress reconstruction.11In the above processes, the blade vibration monitoring and vibration displacement analysis methods are the foundation.If the blade vibration displacement is inaccurate or unreliable,the subsequent parameter identification and application of blade vibration will become meaningless. For example,High Cycle Fatigue (HCF) science and technology program12proposed that the blade stress reconstruction error based on the BTT method should be within 20%. If the error of the blade vibration displacement analysis is large, the blade stress reconstruction error will easily exceed 20%, and the blade vibration stress reconstruction will be invalid.

Among the existing methods of blade vibration monitoring and vibration displacement analysis based on the BTT method, the main method is the Traditional BTT (T-BTT)method with the Once-Per-Revolution (OPR), which uses the OPR as the reference time and rotation speed calculation without vibration. In order to solve the problem that the calculation error of blade vibration displacement caused by using the OPR as the reference is large when the rotation speed is unstable, Carrington6proposed to interpolate the same reference times as the number of blades between adjacent OPR times to calculate the blade vibration displacement,but the calculation error of the blade vibration displacement at the sudden rotation speed is still not guaranteed. Zhang et al.13proposed the multi-reference phases to calculate the blade vibration displacement. Although the calculation error can be reduced to a certain extent, the increase of the number of the reference phases will make the failure of the reference phases easier to occur, which will lead to more time spent to eliminate the error signals and make the vibration displacement calculation invalid. Diamond et al.14proposed that the shaft instantaneous angular speed and instantaneous angular position can be derived by using a state space model with a Kalman filter, and the proposed method can be used to analyze the blade vibration displacement under transient conditions. Ren et al.15proposed the Error Correction BTT (ECBTT) method based on the analysis of the calculation errors caused by the unstable rotation speed,which reduced the inaccuracy of blade vibration displacement calculation obtained by the T-BTT method.

In the confined space of aero-engine, there are situations where the OPR probe cannot be installed or the OPR signal fails. In order to adapt to the blade vibration displacement analysis under the OPR-free condition, the OPR-free BTT methods are proposed to monitor and analyze the blade vibration displacement.The OPR-free BTT methods can be divided into three categories.Firstly,the ToA of the blade obtained by one probe is used as a reference to calculate the vibration displacement of the blade when it reaches another probe.16-18Secondly, the ToA of one blade is used as the reference to calculate the vibration displacements of other blades.19Thirdly, the reference time is obtained by linear fitting of the actual arrival time of the whole-circle blades to calculate the blade vibration displacement.20-22The linear fitting BTT method was first proposed by Russhard,who obtained the fitted ToA by linearly fitting the angles and actual arrival time of the whole-circle blades under only one blade tip probe, and selected the first blade’s fitted ToA as the reference to calculate the vibration displacements of all blades.20On the basis of the linear fitting method of Russhard, Chen et al.21proposed the Compound Reference BTT(CR-BTT)method,and the middle blade was selected as the reference to calculate the blade vibration displacement, which improved the accuracy of the blade vibration displacement analysis. Fan et al.22proposed the shifting straight-line fitting method on the basis of the CRBTT method to analyze blade vibration displacement. It is to ensure that each blade is the middle blade by translation point by point.But this will greatly increase the computing time.He et al.23proposed an improved technique to reduce such random errors by substituting the real key-phase signal with a reconstruction by means of the tip timing signal. When using the linear fitting method to analyze the blade vibration displacement, there are still two problems to be explored: one is whether the blade vibration characteristics will affect the fitting results,and the other is how to suppress the error since the linear fitting at variable speeds will inevitably lead to calculation errors.

In this paper, starting with the linear fitting of the ToA of the rotor blades without vibration at constant rotation speed,the expression of the Theoretical Arrival Time (TAT) is derived. On the basis of ToA without vibration, the time difference caused by vibration is introduced to deduce whether the TAT of blade can be accurately restored by the linear fitting method when the rotor blades vibrate. Combined with the vibration characteristics of the whole-circle blades, it can be known that the TAT obtained by linear fitting with a single probe will deviate from the actual TAT when the rotor blades vibrate synchronously. This will lead to the deviation of blade vibration displacement from the actual value.In order to solve the shortcomings of linear fitting with a single probe, the Multi-Probe linear fitting method based on the BTT (MPBTT)is proposed and the theoretical derivation is given.Since the blade vibration displacements captured by different probes are inconsistent (positive and negative), the fitting deviation caused by synchronous vibration can be suppressed, so as to improve the accuracy of TAT of the blade obtained by linear fitting method. The linear correction of ToA method is proposed to reduce the problem of linear fitting error caused by the nonlinear change of rotation speed. Combined with the MP-BTT method and the linear correction of ToA method,the MP-LC-BTT method without the OPR is proposed to analyze the blade vibration displacement. Simulation and experimental results verify that the MP-LC-BTT method can improve the accuracy of blade vibration displacement analysis.The analysis idea of the proposed method is not limited to the linear fitting, but includes polynomial fitting to improve the accuracy of blade vibration displacement analysis.

2. Theory of multi-probe linear fitting without OPR

2.1. Principle of blade vibration displacement analysis based on BTT method

There have been many articles on the principle of blade vibration displacement analysis based on the BTT method.6,7,15The following is a brief description about it.The BTT method is to capture the difference between the actual ToA when the blade reaches the tip probe and the TAT without vibration,and then calculate the vibration displacement of the blade by combining the blade tip linear velocity. In the actual measurement process, the actual time of the blade reaching the probe can be measured,but the TAT without vibration cannot be measured.The blade vibration displacement can only be obtained indirectly by other methods. The main solution at present is to set a key phase on the shaft or disk as the OPR (required to be no-vibration), which is used as the reference point of the rotor blades. The blade vibration is obtained by calculating the change of the arc angle between each blade and the OPR. If the blade does not vibrate, the arc angle between the blade and the OPR is fixed, which can be called the theoretical arc angle; if the blade vibrates, the actual arc angle between the blade and the OPR will deviate from the theoretical arc angle without vibration.The vibration displacement of the blade can be obtained by calculating the difference between the actual arc angle and theoretical arc angle and multiplying by its radius. The schematic diagram of blade vibration displacement measurement based on the BTT method is shown in Fig. 1, where alr,brepresents the real arc length between the OPR and the Blade #b, which may have vibration; alm,brepresents the theoretical arc length when the Blade #b has no vibration;tbrepresents the ToA of the Blade#b with vibration; topris the ToA of OPR.

There are three main purposes for introducing the OPR:(A) As the reference point without vibration to calculate the ToA of each blade relative to the key phase; (B) get the rotation speed of the rotor blade;(C)identify the physical number of the rotor blades.

In order to avoid the failure to calculate the blade vibration displacement due to the loss or error of the OPR signal, the BTT methods without OPR were proposed. Many of the current BTT methods without OPR,especially based on linear fitting,still need the reference time. But the reference time is not generated by a physical mark,which obtained by the linear fitting with the actual ToA of whole-circle blades. However, in order to calculate the blade vibration displacement accurately,the reference time obtained by linear fitting should have the same characteristics as the time of physical OPR. That is,the fitted reference time is not affected by the blade vibration,or the influence is small.

Therefore, the key point of the BTT method without OPR based on linear fitting is how to accurately obtain the TAT of the blade without vibration. Furthermore, when using linear fitting to obtain the TAT, we need to know how to ensure the accuracy of the linear fitting, that is, when the ToA of the whole-circle blades and their position angles do not change in a linear trend,how to ensure the accuracy of blade vibration displacement analysis, such as acceleration or deceleration process.

For the above problems, this paper starts with the existing single probe linear fitting BTT method to deduce the accuracy of the TAT.Then the proposed multi-probe linear fitting BTT method is introduced to deduce its effectiveness.

2.2. Principle derivation of linear fitting BTT method without OPR

2.2.1. Definition of relationship model between ToA of wholecircle blades and their position angles

Fig. 1 Schematic diagram of blade vibration displacement measurement based on BTT method.

The model of the mathematical relationship between the ToA of the whole-circle blades and their position angles in a single revolution is related to the rotation speed. When the rotation speed of the rotor blades is constant,the relationship model of the ToA and the blade position angles are linear. When the rotation speed of the rotor blades is accelerated or decelerated,the relationship between them becomes more complex,and the relationship model can be obtained according to the derivation. In this paper, the linear fitting BTT method is used to analyze the blade vibration,so the relationship model between the ToA and the position angles of the whole-circle blades is defined as linear, which can be expressed as.

where φ1brepresents the angle of Blade#b relative to Blade#1 when Blade#1 is defined as 0°,and the angle value range is(0,2π);b is the blade index,and its range is from 1 to B,which is the total number of the whole-circle blades;tmodel，bis the actual ToA of Blade#b;lmodelis the slope of fitted line obtained by the least square linear fitting of the position angles of whole-circle blades and their actual ToA; bmodelrepresents the intercept of the fitted line; τmodelis the difference between the actual ToA of different blades and the time obtained by linear fitting.For the different operation processes,the corresponding value of each blade is also different.

In the possible operation process of the rotor blades, the constant rotation speed conforms to the process of linear change between the ToA and the position angles of the whole-circle blades. The following is the theoretical analysis of the linear fitting method without OPR of the constant rotation speed process, which is then extended to other complex operation processes.

2.2.2. Relationship between position angles and ToA of wholecircle blades under constant rotation speed without vibration

The ideal constant rotation speed of the rotor blades does not appear in the actual operation, but is accompanied by small random fluctuations. Thus the real constant rotation speed can be expressed as.

where tTAT，bis the time when Blade#b reaches the probe without vibration, that is, the TAT, unit: s. The relationship between TAT tTAT，band relative angle φ1bof each blade can be obtained by combining Eq. (2) and Eq. (3):

where τe（φ1b） is the time fluctuation caused by the random fluctuation of the rotation speed. By comparing Eq. (1) and Eq. (5), we can get τe（φ1b）=τmodel（φ1b）.

When the least squares linear fitting method is used to fit the ToA and position angles of the whole-circle blades under the constant rotation speed without vibration, the obtained expression is defined as.

Combining Eqs. (5)-(8), the difference between the TAT and fitted ToA of Blade #b obtained by linear fitting under the constant rotation speed without vibration is.

When the value of τe（φ1b） is small, for example, the constant rotation speed contains small random fluctuations, the effect on the accuracy of the fitting is mainly determined by.

!

2.2.3. Relationship between position angles and ToA of wholecircle blades under constant rotation speed with vibration

The arrival time of blade reaching the monitoring probe will have a slight deviation from the TAT when the blade vibrates.The difference in arrival time of each blade due to vibration is represented by Δtv，b, and the actual arrival time of each blade with vibration can be expressed as.

Therefore,the process of using the actual arrival time of the whole-circle blades combined with their position angles to fit the TAT of the blade can be expressed as.

Since the Linear Fitting (LF) of the whole-circle blades is performed to obtain the TAT, it is necessary to comprehensively consider the impact of the vibration characteristics of the whole-circle blades on the fitting to ensure that the analysis is accurate and meaningful, which has not been considered in the previous studies.

There are two kinds of regular vibration of rotor blades,one is synchronous vibration, and the other is asynchronous vibration.For the whole-circle blades with synchronous vibration occurring,the vibration trends of the blades obtained one by one through monitoring probe are consistent.If the wholecircle blades are perfectly tuned, the vibration displacement of each blade monitored by one probe is the same. If the wholecircle blades are mistuned, the vibration trend of each blade monitored by one probe is consistent,but the center frequency and amplitude of each blade are respectively different. Fig. 2 shows the vibration trends of the whole-circle blades obtained through the experimental analysis.

When the rotor blades vibrate asynchronously, the vibration trend of the whole-circle blades is different,and the vibration of each blade fluctuates according to the law. According to the vibration characteristics of the whole-circle blades, the accuracy of the TAT of the blade obtained by linear fitting method is separately analyzed.

(1) Linear fitting of synchronous vibration of rotor blades

When the rotor blades are perfectly tuned, which does not happen in practice, the vibration displacement of each blade monitored by the same probe is the same. That is, the arrival time difference Δtv，bcaused by the vibration of each blade is the same, which is expressed by Δtt. Then Eq. (12) can be expressed as.

tr，b=tTAT，b+Δtt（14）

If the actual arrival time monitored by a single probe is used for linear fitting,the parameters of the fitted line are given as follows:

As can be seen from Eq.(15),when the whole-circle blades are perfectly tuned,the actual arrival time with vibration is linearly fitted to obtain the slope of the fitted line consistent with that without vibration. It can be seen from Eq. (16) that the intercept of the fitted line with vibration is inconsistent with that without vibration,and the fitted arrival time of each blade includes the same time deviation Δttcaused by synchronous vibration.It is invalid to use the actual arrival time monitored by only a single probe to perform linear fitting to obtain the TAT.If the point on the fitted line is used as the reference time to calculate the vibration displacement of each blade which is also 0,the actual vibration of the whole-circle blades cannot be restored.

The arrival time difference Δtv，bof each blade due to vibration is different when the whole-circle blades are mistuned.Because the vibration trend of each blade is the same,the arrival time difference Δtv，bis usually the same sign. The slope of the fitted line obtained by linear fitting remains unchanged,and the intercept of the fitted line will change, as follows:

Because Δtv，bis the same sign, the fitted time must deviate from the TAT. If the fitted value is used as the reference time to calculate the blade vibration displacement,it will inevitably contain a certain deviation.

Fig. 2 Vibration trends of whole-circle blades obtained through experimental analysis.

(2) Linear fitting of asynchronous vibration of rotor blades

The vibration frequency of each blade is the same when the whole-circle blades vibrate asynchronously. And the vibration phase of each blade is continuous in one revolution.The vibration displacements of the whole-circle blades monitored by the same probe are the points which distribute on the sine function line, as shown in Fig. 3.

The arrival time difference Δtv，bcaused by blade vibration in Eq. (12) under asynchronous vibration can be expressed as.

where Δtvis the amplitude of time difference caused by asynchronous vibration, which can be considered as the same for the whole-circle blades; Ωvrepresents the rotational angular velocity of the blade; φv，bis the vibration phase difference when different blades reach the same probe. Substituting Eq.(18) into Eq. (17), we can obtain.

The fitted arrival time obtained by linear fitting of asynchronous vibration can be approximately regarded as the TAT. However, it should be noted that when the number of blades participating in the linear fitting is relatively small, the result of Eq. (19) is not approximately 0. In other words, the fitted arrival time is deviated from the TAT,which will reduce the accuracy of the analysis result of the blade asynchronous vibration displacement.

From the above analysis results,using a single probe linear fitting in synchronous vibration will lead to larger error. And the error of asynchronous vibration analysis cannot be determined when the number of blades is small. In order to make up for the above shortcomings of a single probe linear fitting,the multi-probe linear fitting method is proposed to improve the accuracy of the blade vibration displacement analysis based on the BTT method without OPR.

2.3. Multi-Probe linear fitting method based on Blade Tip Timing (MP-BTT)

Fig. 3 Vibration displacement distribution of whole-circle blades monitored by the same probe during asynchronous vibration.

It can be seen from the previous analysis that the vibration trend of the whole-circle blades monitored by a single probe is consistent when the blades vibrate synchronously.The reference time obtained by linear fitting must contain errors,which make the blade vibration displacement deviate from the actual value.

In the actual measurement, in order to obtain more vibration information when the blades undergo synchronous vibration, it is often required to arrange multiple probes with scattered positions to avoid the same vibration displacement of the blade monitored. The synchronous vibration displacements of the same blade monitored by probes with different arrangement angles are shown in Fig. 4.

In practice, the vibration displacements of the same blade monitored by different probes will not be completely symmetrical.But in most cases,it will be ensured that the value sign of the vibration displacements will be opposite, which provides the possibility to reduce the error caused by the consistent trend of synchronous vibration of the whole-circle blades.The principle of the MP-BTT method is described in detail below.

Fig. 5 is a schematic diagram of 36 blades and 5 blade tip probes. The angle between Probe #1-#5 and Probe #1 in the reverse rotation direction is represented by θ1p, and the subscript p represents the probe index.

The arrival time when Blade #1 reaches Probe #1 is the start time of one revolution, and the position angle is 0°.The absolute position angles of other blades to the different probes can be obtained according to the angles θ1pbetween probes and the angles φ1bbetween blades.Fig.6 shows the relative position of the probes and the blades in Fig. 5 expanded in the circumferential direction.

It can be seen from Fig. 6 that, when the blades are rotating, Probe #1 first detects the arrival time of Blade #1, then Probe #5 detects the arrival time of Blade #29, then Probe#3 captures the arrival time of Blade#6,and the other probes are similar. According to the above-mentioned rotation sequence, each blade will pass through five probes in one revolution, and the absolute position angles of the whole-circle blades corresponding to the five probes distribute in the range of [0, 360°]. And the absolute position angles of the wholecircle blades corresponding to each probe can be calculated according to the angles between the probes and the angles between the blades.

Fig. 4 Synchronous vibration displacements of the same blade monitored by probes with different arrangement angles.

Fig. 5 Schematic diagram of whole-circle blades and blade tip probes.

If the absolute position angle of each blade corresponding to Probe #1 is φ1b, and the absolute position angle of each blade corresponding to Probe #2 is φ1b-θ12. If it is negative,it needs to be converted to the range of[0,360°].And the absolute position angle of each blade corresponding to other probes can be calculated similarly.There is a one-to-one correspondence between the arrival time of each blade captured by each probe and the absolute position angle of each blade corresponding to each probe.φ1b，prepresents the absolute position angle of the blade corresponding to each blade tip probe. According to Eq. (5), the arrival time of the blade without vibration and without the consideration of random error is given by.

The actual arrival time of blade with vibration can be expressed as tr，b，p=tTAT，b，p+Δtv，b，p, and Δtv，b，prepresents the arrival time difference of the blade corresponding to different probes due to vibration. At constant speed, the parameters obtained by the least square linear fitting of the arrival time and the absolute position angles of the multi-probe blades without vibration can be expressed as.

where P is the total number of probes.The detailed derivation of Eq. (22) can be found in Appendix A. Eqs. (22) and (23)show the parameters of fitted line of the TAT without vibration. The fitted line also satisfies the minimum approximation of the TAT when the random time error is included.

The actual arrival time and the absolute position angles of the whole-circle blades corresponding to all blade tip probes with vibration are brought into Eq. (22), and then the slope of fitted line is given as.

The detailed derivation of Eq.(24)can be found in Appendix B. The intercept of fitted line can be obtained by referring to Eq. (23).

By comparing with Eqs. (17) and (25), it can be seen that the signs of the arrival time differences caused by the vibration of the whole-circle blades under the monitoring of multiple probes are different. In general, the intercept of Eq. (25) is close to that of fitted line of the TAT without vibration. For asynchronous vibration,the fitted line obtained from the blade arrival time of multiple probes has more data involved in the calculation, and the fitted time is closer to the TAT without vibration, which will improve the accuracy of blade vibration displacement analysis.

Fig. 6 Schematic diagram of the relative position of blade tip probes and blades after unfolding along running direction.

In the above analysis,it is assumed that the rotor blades are running at a constant speed. But the relationship model between the arrival time and the position angles of the wholecircle blades is not linear when the rotation speed is variable.It will inevitably lead to data fitting errors when the linear fitting method is used. In order to reduce the fitting errors caused by variable rotation speed, a Linear Correction (LC) method of ToA is proposed in this paper, and the details are shown below.

3.LC of ToA of whole-circle blades with the nonlinear change of rotation speed

3.1. Derivation of LC method

In the existing researches,15,20it is found that the process of the nonlinear change of rotation speed can be approximately as a linear change process within a single revolution.Based on this,the real-time rotation speed of the rotor blades can be expressed as.

where Ω0represents the start rotation speed of the linear change process, unit: Hz; k is the change rate of the rotation speed, unit: Hz/s. The relationship between the position angle and the ToA of the blade without vibration is shown as.

where t1is the start time of the linear change process within a single revolution. Note that in order to simplify the analysis and representation, the ToA and the position angles of the whole-circle blades corresponding to only single probe are used for analysis, which is the same for multiple probes.

Substituting Eq. (26) into Eq. (27) yields.

Since the rotation speed,in Eq.(31),at the initial time of a single revolution cannot be accurately obtained, the rotation speed of each blade cannot be calculated. However, the average rotation speed Ωmeanof the whole-circle blades can be obtained. The average rotation speed Ωmeanis approximated by the rotation speed Ωmat the average position angle of the whole-circle blades. In fact, there is a certain error between them, which is shown by.

The larger the rotation speed of the blade is, the closer the average rotation speed is to the rotation speed at the average position angle, and the smaller the error is. The error between them is large when the rotation speed is small.

Taking the average rotation speed as the reference,the rotation speed of each blade can be expressed as.

It can be seen from Eq.(37)that when the rotor blades are running at a linear change speed without vibration,the slope m of the fitted line is related to the position angle φband the relative time difference Δtm，bof each blade, and the relative time difference is related to the average rotation speed Ωmeanof the whole-circle blades and the change rate k of speed in a single revolution.Thus when we knowφb,Ωmandk,the slope of fitted line can be calculated. The above three quantities can be obtained by calibration and real-time analysis.

The intercept c of the fitted line can be expressed as.

Put the slope and the intercept obtained by Eqs. (37) and(38) into the objective formula of the fitted line, and the fitted ToA of each blade is shown as.

Eq.(40)is similar to Eq.(37),and its value can also be calculated. Adding the fitting error τbto the fitted ToA can correct the ToA ^tb，LC, which is closer to the TAT of blade without vibration. The accuracy of blade vibration displacement analysis is improved by reducing the calculation error caused by linear fitting under the nonlinear change of rotation speed.

Fig. 7 is the comparison between the fitted ToA and real ToA at different rotation speeds with the same speed change rate and the comparison between the time error caused by linear fitting and the corrected time error calculated by Eq. (40).The average rotation speed is used to replace the corresponding rotation speed at the average position angle of the whole-circle blades to calculate the fitted time difference obtained by Eq.(40),which is close to the actual fitted time difference when the rotation speed of the blade is large,as shown in Fig. 7 (a). When the rotation speed is small, the error between the average rotation speed and the rotation speed corresponding to the average position angle is relatively large,which will result in a large error in the correction time far from the middle position,as shown in Fig.7(b).But this is only the error of correction time,and the error of blade vibration analysis is still attributed to the vibration displacement. The following is an analysis of the blade vibration displacement in combination with the rotation speed.Since the radius of blade is unknown,the value of vibration radian is used instead of the vibration displacement of the blade, unit: radian.

The real vibration error of each blade caused by the linear fitting is calculated by.

The vibration radian error caused by the linear correction method can be analyzed by Eqs. (42) and (43).

Fig. 8 shows the comparison of blade vibration errors obtained by the linear correction method under the same rotation speed and different speed change rates.

It can be seen from Fig.8(a)that the maximum error after linear correction of ToA is reduced to about 2%of that before correction. The maximum error is reduced to about 25% of that before correction, as shown in Fig. 8 (b). Consistent with the previous analysis, the larger the rotation speed, the better the accuracy of the ToA correction.

Fig.9 is the comparison of the maximum errors before and after linear correction under five conditions of linear speed change rate of 5 Hz/s, 15 Hz/s, 25 Hz/s, 35 Hz/s and 45 Hz/s when the initial rotation speed of blade is 20-60 Hz.

3.2. Relationship between the limit of error of blade vibration and the rotation speed and its change rate

According to Eq. (42), the error at the position angle φb=0 where linear fitting is used without vibration is the maximum,which can be expressed as.

Fig. 7 Comparison of fitted ToA and real ToA and comparison of fitting error and corrected error.

Fig. 8 Comparison of errors caused by linear correction method under different rotation speeds.

Fig. 9 Comparison of maximum errors before and after linear correction under different k and different initial rotation speeds.

The maximum speed change rates under different error limits and different rotation speeds can be calculated from Eq.(45). For example, when the blade tip radius is 100 mm, the error limits are 10 μm, 20 μm, 30 μm, 40 μm, and 50 μm,and the maximum speed change rates obtained by Eq. (45)at different rotation speeds are shown in Fig. 10.

It can be seen from Fig.10 that when the speed change rate is 1 Hz/s and the error limit is 1 × 10-4, the minimum allowable rotation speed directly using linear fitting is 70 Hz.At the same speed change rate,the minimum allowable rotation speed can be reduced when the error limit increases.If the error limit is 5 × 10-4, the minimum allowable rotation speed is 32 Hz.

By combining with Eqs. (35) and(43),the maximum speed change rate corresponding to different rotation speeds under different error limits after linear correction of ToA of the blade can also be analyzed. And the corrected maximum vibration radian error in a single revolution is shown by.

Fig.11 shows the maximum speed change rates corresponding to different rotation speeds under different error limits after liner correction of ToA.The rotation speed range is from 5 to 70 Hz,and the maximum value of the speed change rate is 100 Hz/s.Because it is quite different from the actual operating conditions, the value of the speed change rate exceeding 100 Hz/s will not be shown here.

It can be seen from Fig. 11 that when the rotation speed is 5 Hz and the error limit is 1 × 10-4, the maximum allowable speed change rate is no more than 1 Hz/s;the maximum value is more than 100 Hz/s under the same error limit when the rotation speed exceeds 70 Hz.

By comparing Fig.10 and Fig.11,it can be known that the maximum allowable speed change rate under the same error limit is greatly increased after the linear correction of ToA,which also indicates that the linear correction method is effective and can improve the accuracy of the blade analysis results.

4.Multi-Probe linear fitting and ToA Linear Correction method based on BTT (MP-LC-BTT)

Through the above analysis, the MP-LC-BTT method first obtains the TAT of blade by multi-probe linear fitting, and then is combined with the linear correction of ToA to obtain a more accurate TAT of the blade during the nonlinear change of rotation speed. Then the blade vibration displacement can be calculated by the relationship between the slope of the fitted line and the rotation speed of each blade.

In the actual measurement,it is firstly necessary to calibrate the angles between the whole-circle blades and the angles between the probes, and then determine the absolute position angles of the whole-circle blades under different probes,which can be seen in Section 2.3. According to the calibration of blade position angles in Ref. [15], the blade position angles can be calculated by recording the ToA of the whole-circle blades B + 1 to the range of [0,360°] when they are running at a constant speed. Other probes can do the same process to get the average angles between the blades to reduce the error.

Fig. 10 Maximum speed change rates of different rotation speeds under different error limits obtained by linear fitting method.

Fig. 11 Maximum speed change rates of different rotation speeds under different error limits obtained by linear correction method.

The blade vibration displacement analysis based on MPLC-BTT method is carried out as follows.

The slope ln，t，fand the intercept bn，t，fof the fitted line under one revolution are obtained by least square linear fitting of the absolute position angles and the corresponding ToA of the whole-circle blades under multiple probes. Then the TAT of Blade #b corresponding to blade tip Probe #p obtained by the linear fitting method can be expressed as.

where the subscript n, TAT, b, p, and f respectively represent revolution number,theoretical arrival time,blade index,probe index, and fitting.

It can be known from Eqs. (24) and (37) that the relationship between the slope obtained by linear fitting and the average rotation speed is approximately shown as.

The speed change rate kncan be obtained by calculating the rotation speed and ToA at the average angle of two adjacent revolutions.

When the speed change rate knis 0, that is, the rotation speed is constant, there is no need for linear correction of the TAT, which obtained by linear fitting can be regarded as the reference time of the blade without vibration.

When the speed change rate knis not 0,the average rotation speed Ωn，meanand the speed change rate knobtained by Eqs.(48)and(50)are brought into Eqs.(35)and(40)to get the time difference τn，b，pbetween the fitted ToA and the TAT without vibration. According to Eq. (41), the corrected TAT tn，TAT，b，p，cunder the nonlinear change of rotation speed is shown as.

Whether the calculation accuracy will be affected when the speed change rate knis close to 0 can be seen in Appendix C.

Fig. 12 is the flowchart of blade vibration displacement analysis based on MP-LC-BTT method.

In order to improve the calculation efficiency and accuracy,two tips are briefly introduced in the analysis process.

(2) The installation angles of probes can be determined according to the distribution requirements of the parameter identification method. If one wants to further improve the accuracy of the measurement, one can do further research on it. Due to the length of this article,there are no more details here. The probes layout requirements of parameter identification method are adopted by default.

Fig. 12 Flowchart of blade vibration displacement analysis based on MP-LC-BTT method.

5. Simulation verification of MP-LC-BTT method

5.1. Introduction of simulation model

5.1.1. Simplified excitation forces of whole-circle blades

In the existing simulation models,only the harmonic excitation force is applied,but the relationship between the actual excitation forces and the whole-circle blades is not considered. Ref.[24] simplified the uneven forces of the flow field on the blade to the Periodic Rectangular Pulse Wave Forces (PRPWF),which are more consistent with the actual excitation forces on the blade. The following is a brief description of the PRPWF on the blade and the relationship between each blade and each excitation force.

As shown in the left of Fig. 13, the PRPWF and the initial position of the whole-circle blades are placed on the same cross-section.The right of Fig.13 shows the position relationship between the whole-circle blades and the PRPWF after being expanded in the circumferential direction. At the initial time, the phase difference between each blade and the excitation force at different positions is different.

According to the above analysis, the excitation force of each blade can be expressed as.

where NFis the number of PRPWF;Fiis the amplitude of each PRPWF, unit: N; square(fre, duty) represents the function of PRPWF, the parameter fre represents the time series, and the parameter duty represents the duty of each PRPWF; φ0is the initial phase; φiis the phase of each PRPWF; φbis the phase difference between each blade and PRPWF; diis the duty of each PRPWF. If the PRPWF and the whole-circle blades are uniformly distributed, we have.

Fig. 13 Schematic diagram of position relationship between whole-circle blades and excitation forces.

Different from the previous parameter setting of the Engine Order (EO) to get synchronous vibration, the synchronous vibration and its EO are realized by setting the number of excitation forces.The relationship between the EO of synchronous vibration and the number of the excitation forces can be seen in Ref. [24].

5.1.2. Parameters of simulation model

The parameters of simulation model are divided into three parts: parameters of rotor blades, parameters of excitation forces, and angles of probes, as shown in Table 1.

In fact,the rotor blades are mistuned,but the degree of mistuning is different.A certain random range of mass is added to the basic mass to characterize the mistuning of the rotor blades.According to the mass,stiffness and damping of blade,the natural frequency of it can be calculated. The whole-circle blades and the excitation forces of the simulation model are uniformly distributed.

Table 1 Parameters of simulation model.

Fig. 14 shows the blade vibration displacement monitored by a probe under the excitation forces in Table 1 when the rotation speed is 0-110 Hz. It can be found that the synchronous vibration of the blade occurs only when the value of EO is an integral multiple of the number of excitation forces 5,which is consistent with the actual situation,and verifies the reliability of the simulation model.

When the rotor blades are mistuned, because the frequencies of the resonance of the blades are close, the phase difference caused by the time difference of resonance has little effect on the vibration trend of each blade, as shown in Fig. 15 (a). It can be considered that the vibration trend of each blade is the same. Fig. 15 (b) shows the vibration trends of the whole-circle blades under an EO measured in the experiment. In order to facilitate the comparison of the vibration trend of each blade,the analyzed blade vibration displacement is filtered.The comparison in Fig.15 also verifies the reliability of the simulation model.

5.2. Results of simulation

5.2.1. Blades are perfectly tuned

In order to verify the reliability of the MP-LC-BTT method,the analysis results of two synchronous vibration points are compared. The first group: the rotation speed of the wholecircle blades is 74.5-77.5 Hz(EO is 30),and the speed-up time is 10 s;the second group:the rotation speed is 89-94 Hz(EO is 25),and the speed-up time is 5 s.The parameters of the blade,excitation forces and probes are shown in Table1. The single probe CR-BTT method and the MP-LC-BTT method are compared to analyze the vibration displacement of a blade with four probes, as shown in Fig. 16 and Fig. 17.

According to the results from Fig.16 and Fig.17,when the blades are perfectly tuned, the synchronous vibration of the blade cannot be restored by linear fitting with only a single probe, which is consistent with the previous deduction and analysis conclusion. However, the synchronous vibration of blade can be restored to a certain extent when the MP-LCBTT method is used. Because the sum of time difference of ToA captured by all probes cannot be guaranteed to be 0,there will be some errors in the analysis of vibration displacement.

Fig. 14 Blade vibration displacement monitored by a probe under the parameters in Table 1.

Although it is impossible for the blade to be tuned in practice, if the blade that is mistuned is small, the fitting error of single probe is larger than that of multiple probes.

5.2.2. Blades are mistuned

In the above operation section,the maximum mistuned degree of the blade is set as 2%,the results of linear fitting with single probe and multiple probes are compared, and the Average Absolute Deviations (AAD) of the whole-circle blades under different analysis methods are analyzed.

From the analysis results of Figs.18-21,it can be seen that when the blades are mistuned, because the vibration displacements of the whole-circle blades are different under the same revolution, the linear fitting with a single probe can also be used to analyze the vibration of the blade.However,compared with the fitting results of the CR-BTT method,the AAD of the fitting results of the MP-LC-BTT method is reduced by more than 65%. The results of MP-LC-BTT method are closer to the actual vibration of the blades, and the vibration trends of the blades are also in good agreement, which can ensure the accuracy of the further phase analysis of the synchronous vibration.

5.2.3. Comparison of the results of linear correction of ToA

Fig. 22 shows a comparison of the blade vibration displacement before and after the linear correction of ToA.

Since the speed change rate is 4.4 Hz/s, it can be seen that the blade vibration displacements obtained by the MP-LCBTT method are very accurate when the rotation speed exceeds 15 Hz in Fig.22(a),which are consistent with the analysis results in Fig.11.Because the blade in Fig.22(b)is in the middle, the linear correction results are very accurate, which are consistent with the analysis results in Fig. 8.

From the above simulation,it can be known that the linear correction of ToA can better correct the error caused by the nonlinear change of rotation speed.

6. Experiment

Due to the limitation of the test-bench conditions,the MP-LCBTT method was used to verify the fitting accuracy and correction accuracy of blade vibration displacement with the nonlinear change of rotation speed on two high-speed straight blade test-benches. Finally, the reliability and accuracy of the MPLC-BTT method were verified on an industrial axial fan. The following is a detailed introduction.

6.1. Experiment on high-speed straight blade test-bench 1

The test-bench 1 arranges four blade tip probes at 6° intervals to monitor blade vibration.The detailed structural parameters and test instruments of the test-bench can be found in Ref.[25].

Fig. 23 shows the vibration displacements of three blades obtained by the T-BTT method. And the blade vibration displacements are processed by Savitzky-Golay (SG) filter. It can be seen from Fig. 23 that the filtered blade vibration displacements still have calculation errors due to the instability of the OPR, such as many burrs marked in the figure. If the burr occurs at the harmonic resonance point, it will affect the analysis of the actual vibration amplitude of the blade.

Fig. 24 shows the vibration displacement interception of Blades #1 and #17 monitored by Probe #1. Due to the blade mistuned, the blade vibration displacement obtained by the CR-BTT method has fluctuation errors, which will affect the analysis of coupled vibration, as shown in Fig. 24 (a).Fig. 24 (b) shows that the blade vibration displacement analyzed by the CR-BTT method is smaller than the actual vibration displacement by more than 20 μm.This error will directly affect the accuracy of blade stress measurement when the blade stress is reconstructed by the blade vibration displacement.The blade vibration displacement obtained by the MP-LCBTT method is in good agreement with the actual vibration displacement obtained by the T-BTT method,and the calculation errors of the vibration displacement caused by the instability of the OPR will not occur. The experimental results verify that the blade vibration displacements obtained by the MP-LC-BTT method are accurate and reliable.

Fig. 15 Vibration trends of whole-circle blades at synchronous vibration point obtained by simulation and experiment.

Fig. 16 Comparison of the results of two fitting methods when rotation speed is 74.5-77.5 Hz.

Fig. 17 Comparison of the results of two fitting methods when rotation speed is 89-94 Hz.

Fig. 25 shows the comparison of the AAD of the vibration displacement of the whole-circle blades obtained by the CR-BTT method and the MP-LC-BTT method. In order to reduce the influence of the stable section on the AAD,the data of four hundred revolutions before and after the resonance center point of each blade is used to participate in the calculation to ensure that the complete synchronous vibration displacement section is included. By comparing the results, it can be found that the results obtained by the MP-LC-BTT method are more stable and its calculation errors are smaller.The AAD of the fitting results of the MP-LC-BTT method is about 40% less than that of the CR-BTT method.

Fig. 18 Comparison of the results of two fitting methods when rotation speed is 74.5-77.5 Hz (Blade #36).

Fig. 19 Comparison of the AAD of two fitting methods when rotation speed is 74.5-77.5 Hz.

Fig. 20 Comparison of the results of two fitting methods when rotation speed is 89-94 Hz (Blade #1).

Fig. 21 Comparison of the AAD of two fitting methods when rotation speed is 89-94 Hz.

Fig. 22 Vibration displacement before and after linear correction of ToA monitored by Probe #1.

Fig. 23 Blade vibration displacement obtained by T-BTT method.

Fig. 24 Comparison of blade vibration displacement obtained by three different BTT methods.

Fig. 25 Comparison of AADs of whole-circle blades obtained by two fitting methods.

6.2. Experiment on high-speed straight blade test-bench 2

The test-bench 2 and the blade vibration monitoring system are shown in Fig. 26. The number of the whole-circle blades is 32 and its diameter is 140 mm. The excitation forces on the blades are generated by four rows of magnets evenly distributing in the circumferential direction. According to the requirements of synchronous vibration identification, five blade tip probes and one OPR probe are arranged. Since one of the blade tip probes and the OPR probe have signal instability during the measurement, the effective probes are four blade tip probes.

In the measurement process, the slop of the rotation speed change is 0.5 Hz/s. The vibration displacements of the three Blades #1, #17 and #32 before and after using the MP-LCBTT method are respectively compared, as shown in Fig. 27.It can be found that there are some deviations of blade vibration displacement obtained by the MP-BTT method due to the influence of speed-up rate in the low rotation speed section.And the MP-LC-BTT method is used to correct the vibration displacement of the blade,which is consistent with the simulation results in Fig. 22.

Fig. 26 High-speed straight blade test-bench and its vibration monitoring system.

Fig. 27 Vibration displacement before and after linear correction of ToA.

6.3. Experiment on industrial axial fan

The industrial axial fan is described in Refs.[19,21,26],and its main parameters are introduced below. The number of the rotor blades is 36,and its diameter is about 1.7 m.The number of stators in the front stage of the rotor blades is 31. The inlet angles of attack of the rotor blades can be adjusted through the transmission mechanism at the root of the blade to change the air flow. Eight optical fiber probes are arranged in two rows on the casing of the rotor blades,and the circumferential included angle of adjacent probes is 6°.The experimental data is generated by 3 effective probes in one row. A reflector is arranged on the rotor shaft as the OPR mark,and its monitoring probe is also optical fiber probe. The industrial axial fan and the arrangements of probes are shown in Fig. 28.

Due to some sudden changes in the blade rotation speed,the blade vibration displacement obtained by the T-BTT method will contain the calculation errors, so we use the ECBTT method based on the T-BTT method to analyze the blade vibration displacement.

Fig. 29 shows the comparison of the vibration displacements of Blade #1 and #18 obtained by the EC-BTT method,the MP-BTT method and the MP-LC-BTT method. It can be found,through the comparison in the figure,that the MP-LCBTT method is effective, and the calculation errors of vibration displacement caused by the unstable rotation speed are well suppressed, and the accuracy of the blade vibration displacement analysis is guaranteed.

Since the MP-LC-BTT method uses the relative included angles of the rotor blades to calculate the blade vibration displacement, the constant deviation of the blade vibration displacement does not change much when the inlet angles of attack of the whole-circle blades change. However, the constant deviation of blade vibration displacement caused by the change of inlet angles of attack is larger when the EC-BTT method with the OPR is used to calculate the blade vibration displacement. The MP-LC-BTT method needs to be further explored to calculate the blade twist angle when we pay attention to it.

Fig. 28 Industrial axial fan and the arrangement of probes.

Fig. 29 Comparison of the correction of blade vibration displacement calculated by MP-LC-BTT method when rotation speed is unstable.

Fig.30 is a synchronous vibration segment that occurred in the revolution number of 1400-2100 from Fig. 29. Fig. 30 shows the comparison of the vibration displacements of Blade#18 obtained by the MP-LC-BTT method, the EC-BTT method and the CR-BTT method under different probes. In order to facilitate clear comparison, the obtained blade vibration displacements have been processed by the SG filter. The ToA of the blade vibration displacement obtained by the CRBTT method has also been corrected to eliminate the influence of the unstable rotation speed. Because of the change of the inlet angle of attack,the blade vibration displacement obtained by the EC-BTT method is processed to remove the constant deviation. The blade vibration displacement obtained by the MP-LC-BTT method is more consistent with the vibration displacement obtained by the EC-BTT method, while the vibration displacement obtained by the CR-BTT method has some fitting errors, which will affect the accuracy of the blade vibration parameters identification.

7. Conclusions

This paper analyzes the inherent defects of inaccurate linear fitting of TAT due to the vibration characteristics of the whole-circle blades from the actual arrival time through the least squares linear fitting method. The MP-BTT method is proposed to reduce the error of the TAT obtained by the single probe fitting method. From the perspective of the theoretical derivation,it is verified that the proposed method can improve the accuracy of the TAT of synchronous vibration.

The linear correction method is proposed to reduce the linear fitting error of ToA with the nonlinear change of rotation speed.Combined with the available parameters of actual monitoring, the detailed derivation formulas are given to improve the accuracy of blade vibration displacement analysis under the rotation speed sudden changes. Combining the MP-BTT method and the linear correction method, the MP-LC-BTT method is summarized to calculate the blade vibration displacement without the OPR. And the flowchart suitable for actual measurement is given.

Fig. 30 Comparison of vibration displacement of Blade #18 obtained by different BTT methods under different probes.

According to the excitation force characteristics of the whole-circle blades during operation, the blade vibration simulation model is redefined to achieve more realistic blade vibration characteristics and provide more reliable data for simulation verification. The simulation data verifies that the MP-LC-BTT can improve the accuracy of blade vibration displacement analysis, and reduce the calculation errors of blade vibration displacement caused by the unstable rotation speed.

Through the data analysis and comparison of two highspeed straight blade test-benches and an industrial axial fan,the analysis results of the MP-LC-BTT method are consistent with those of BTT method with the OPR, which verifies the reliability and accuracy of the MP-LC-BTT method. The MP-LC-BTT method avoids the calculation errors of the blade vibration displacement caused by the instability of the OPR,and can be used as an alternative to the BTT method with the OPR to realize the blade vibration monitoring in the confined space of the aero-engine.

In this paper,there is no in-depth study on the influence of probes’ arrangement on the accuracy of blade vibration displacement analysis. In the follow-up study, further investigations can be made on the requirements of probes’arrangement for blade vibration parameters identification, so as to improve the accuracy of blade vibration displacement analysis.For the analysis of blade vibration displacement,further analysis and verification are still needed to ensure the accuracy and reliability of the method for monitoring all vibration characteristics. This is also what we will explore next.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work has been carried out with the supports of the National Science and Technology Major Project, China(No. 2017-III-0009-0035) and the Major Program of National Natural Science Foundation of China (No. 51790513).

Appendix A.The detailed derivation of Eq. (22) is shown as follows:

Appendix B.The detailed derivation of Eq. (24) is shown as follows:

Appendix C.The influence on the calculation result is analyzed when the change rate of rotation speed is close to 0.

The limit analysis of Eq. (35) is shown as follows:

Eq. (C1) is consistent with the derivation under constant speed, which means that the MP-LC-BTT method can be applied to any change rate of rotation speed.